Problem 12 · 2022 Math Kangaroo
Medium
Counting & Probability
careful-countingcomplementary-counting
Aladdin’s carpet is a square. Along each edge there are two rows of dots (see diagram), and each edge has the same number of dots. How many dots does the carpet have in total?
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Answer: A — 32
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Hint 1 of 3
The dots make two square loops — a big loop on the outside and a smaller loop just inside it.
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Hint 2 of 3
Count the dots on one side of a loop, but be careful: the corner dots belong to two sides, so don't count them twice.
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Hint 3 of 3
Add up the big loop and the small loop.
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Approach: count the big square loop and the small square loop of dots
- The dots make two square loops, one just inside the other, with the same number on every side.
- The big loop has 6 dots along each side; counting around it (corners only once) gives 4 × 6 − 4 = 20 dots.
- The small loop has 4 dots along each side, giving 4 × 4 − 4 = 12 dots.
- Altogether that is 20 + 12 = 32 dots (choice A).
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